This section outlines conditions under which profits, and therefore the wage of skilled workers in the innovation sector, increase in response to an increase in the number of entrepreneurial firmsN. In particular, we show that there exists a thresholdHO∗ allocation of skilled workers to the old economy firms such that
dπ/dN, dω/dNe
Recall from equation (6) that profits π are proportional to HN, implying that dπ/dN = dHN/dN. Furthermore, since
HN =H−HO−N,
we have that
dπ/dN =dHN/dN =−dHO/dN −1.
This expression will be positive if dHO/dN < −1.
To see the conditions under which dHO/dN < −1, recall that the skilled labor market arbitrage condition (15) implicitly definesH0 as a function of N. From (15) we have
£HOtρ +Lρ¤1−ρρ
Substitute for the steady state ABtt in (42) to get
£HOtρ +Lρ¤1−ρρ
Differentiating both sides of the expression with respect to N, it follows that dHO/dN =− (1−α)α2α/(α−1)γηδλ (ηδNλ )γ−1 expression are graphed as shown below in Figure (6).
To the right of HO∗, HN and profits increase with N. As N goes up, skilled workers are reallocated away from old economy firms and the net inflow into the new economy firms is positive (HN rises). In addition, if skilled workers have some bargaining power, as we have argued previously, wageωewill also increase. SinceeωH is constant regardless of the number of entrepreneurial firms (wage result 1 in the main text), asωeincreases within group inequality (measured by the ratio ωeeω
H) will also increase.
However, we know that as N increases, the number of skilled workers in the old economy (HO) falls. Starting from an initial allocation to the right ofH∗,as financial markets improve and N increases, if N becomes sufficiently big (so that HO becomes sufficiently small and is to the left of the threshold HO∗), the economy eventually ends up in the region where HN and profits decrease withN. As we have argued in the main text, data on profits / GDP for the US suggest that for the period of analysis, in terms of our model the economy has been operating to the right of H∗. The model therefore delivers the prediction that within-group inequality increases in response to an improvement in financial markets.
Figure 6: Threshold allocation of skilled workers to the old economy firms (when HO > HO∗, profits rise with an increase inN).
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